Question

If $A,B,C$ are acute positive angles such that $A+B+C=\pi$ and $\cot A\cot B\cot C=k$, then

• A. $k<\frac{1}{9}$
• B. $k>\frac{1}{3}$
• C. $k\le \frac{1}{3\sqrt{3}}$
• D. $k\ge \frac{1}{3\sqrt{3}}$

Answer 1

The correct option is C $k\le \frac{1}{3\sqrt{3}}$

$A+B+C=\pi \Rightarrow \tan A+\tan B+\tan C=\tan A\tan B\tan C$

Now, $A.M.\ge G.M.$
$\Rightarrow \frac{\tan A+\tan B+\tan C}{3}\ge (\tan A\tan B\tan C{)}^{\frac{1}{3}}\Rightarrow \frac{\tan A\tan B\tan C}{3}\ge (\tan A\tan B\tan C{)}^{\frac{1}{3}}\Rightarrow (\tan A\tan B\tan C{)}^{\frac{2}{3}}\ge 3\Rightarrow {\left(\frac{1}{k}\right)}^{\frac{2}{3}}\ge 3\Rightarrow k\le \frac{1}{3\sqrt{3}}$